Krieger’s generator theorem shows that any free invertible ergodic measure preserving action (Y,\mu, S) can be modelled by A^Z (equipped with the shift action) provided the natural entropy constraint is satisfied; we call such systems (here it is A^Z) universal. Along with Tom Meyerovitch, we establish general specification like conditions under which Z^d-dynamical systems are universal. These conditions are general enough to prove that
1) A self-homeomorphism with almost weak specification on a compact metric space (answering a question by Quas and Soo)
2) Proper colourings of the Z^d lattice with more than two colours and the domino tilings of the Z^d lattice (answering a question by Şahin and Robinson)
are universal. Our results also extend to the almost Borel category giving partial answers to some questions by Gao and Jackson.